#6552176

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(10,14)(15,17)(16,21)(22,23)(25,26,31,36,30,34,29,35)(27,37,39,49,40,32,44,47)(28,41,42,48,43,50,45,33)(38,52,55,46,54,56,53,51);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,30,40)(26,32,34,37)(28,38,43,54)(29,44,31,39)(33,46,48,51)(35,49,36,47)(41,56,50,52)(42,55,45,53);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,33)(27,38)(29,42)(30,43)(31,45)(32,46)(34,48)(35,41)(36,50)(37,51)(39,53)(40,54)(44,55)(47,52)(49,56);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23)(26,34)(32,37)(33,48)(35,36)(41,50)(46,51)(47,49)(52,56);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,30,31)(26,35,34,36)(27,39,40,44)(28,42,43,45)(32,47,37,49)(33,41,48,50)(38,53,54,55)(46,52,51,56)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459372);